One Ring to Rule Them All: A Unified Topological Framework for 4D Superconformal Anomalies

TL;DR

This paper introduces a comprehensive topological framework for understanding all 4D superconformal anomalies, extending previous models to include complex polynomial structures that explain different anomaly types.

Contribution

It develops a unified topological approach using constraint ideals and cohomology to describe superconformal anomalies, broadening the scope beyond traditional Chern-Simons formulations.

Findings

Unified description of 4D superconformal anomalies

Identification of richer polynomial structures in anomaly characterization

Explanation of coexistence of different anomaly types in conformal theories

Abstract

We present a unified topological description of anomalies that generalizes the Chern-Simons formulation of Yang-Mills anomalies to encompass all 4-dimensional superconformal anomalies. The key innovation is our characterization of anomalies through the constraint ideal in the polynomial ring of generalized curvatures and connections of the underlying symmetry (super)-Lie algebra. We demonstrate that anomalies in dimension $d$ are captured by the cohomology $H_\delta(W_{d+2})$ of the generalized BRST operator $\delta$ acting on the fermion number $d+2$ component of the constraint ideal $W_{d+2}$. While Yang-Mills anomalies correspond to invariant Chern curvature polynomials (where $W_{d+2}$ reduces to homogeneous curvature polynomials), the constraint ideal for 4D (super)conformal gravity contains additional polynomials mixing curvatures and connections. This richer structure naturally explains the coexistence of both Chern-type ($a$) and non-Chern-type ($c$) anomalies in (super)conformal theories.